On an Elementary Question of the Theory of Manifolds
Some infinities are larger than others — the real numbers can never be listed.
Cantor proved that infinity comes in more than one size — and that the numbers filling the number line are a bigger infinity than the counting numbers, by a trick you could draw on a napkin.
The idea, unpacked
To compare two infinite collections, Cantor asked a child's question: can you pair them up one-for-one with nothing left over? By that test, surprisingly many infinities are the same size — there are exactly as many even numbers as whole numbers, and even as many fractions, because you can line each up against 1, 2, 3, …. The counting infinity, ℵ₀, seemed to swallow everything.
Then Cantor looked at the real numbers — every point on the number line, written as an endless string of digits — and proved they cannot be paired up with the counting numbers at all. Hand him any list claiming to contain them all, and he builds, from the list's own diagonal, a number that differs from your first entry in the first digit, your second in the second, and so on. It is nowhere on your list. So the reals are a strictly larger infinity. There is more than one infinity, and no biggest one.
Where it came from
Georg Cantor, working at the modest University of Halle, founded the theory of infinite sets almost single-handedly in the 1870s and 80s. He first proved the reals uncountable in 1874; the gloriously simple diagonal version came in 1891, presented at the very first meeting of the German Mathematical Society, which he had helped found and led as its first president.
It cost him dearly. The powerful Berlin mathematician Leopold Kronecker, who refused to believe in completed infinities, ridiculed the work, tried to suppress it, and branded Cantor a “corrupter of youth.” Shut out of the career he wanted and attacked for his deepest ideas, Cantor suffered repeated breakdowns and spent his last years in and out of a sanatorium. Vindication came slowly: within a generation his set theory was the bedrock of modern mathematics, and David Hilbert proclaimed that no one would ever expel mathematicians from “the paradise which Cantor has created.”
Why it mattered
Cantor gave mathematics its first rigorous grip on the infinite — not a vague “forever,” but a precise ladder of sizes that can be measured and compared. And the little diagonal manoeuvre at the centre of it turned out to be a master key: the same move, slightly rephrased, later showed that no logical system can prove all truths and that no computer can solve every problem. A question about the sizes of infinity became a question about the limits of reason and computation.
The guest who's on no list
Imagine a hotel claims its guest book lists every possible person, each described by an endless row of yes/no traits. You build one troublemaker like this: take the opposite of guest 1's first trait, the opposite of guest 2's second trait, the opposite of guest 3's third, and so on down the diagonal. The result is a perfectly good description of a person — but he can't be guest 1 (they differ on trait 1), can't be guest 2, can't be anyone on the page. The “complete” list was never complete. Build that guest yourself below.
Where it sits in the story
Galileo had already noticed something odd — that you can pair every whole number with its square, as if a part were as big as the whole — and backed away, calling infinity beyond comparison. Cantor walked straight in and built a theory. What followed his diagonal was both crisis and revolution: applied carelessly it produces Russell's paradox, which forced mathematicians to rebuild set theory on careful axioms. And the same diagonal is the direct ancestor of two later landmarks in this library — Gödel's proof that truth outruns proof (1931) and Turing's proof that some problems no machine can decide (1950). The continuum question Cantor couldn't answer was finally shown to be unanswerable from the standard axioms.
Let m and w be any two mutually exclusive characters, and consider a collection M of elements E = (x₁, x₂, …, x_ν, …) which depend on infinitely many coordinates x₁, x₂, …, x_ν, …, where each of these coordinates is either m or w.
This proof is remarkable not only for its great simplicity, but above all because the principle it follows extends at once to the general theorem that the powers of well-defined manifolds have no maximum — or, what is the same, that beside any given manifold L one can place another, M, of higher power than L.